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Electric Dipole Approximation
In general, the wavelength of the type of electromagnetic
radiation which induces, or is emitted during, transitions between different
atomic energy levels is much larger than the typical size of an atom.
Thus,

(1109) 
can be approximated by its first term, unity. This approach is
known as the electric dipole approximation.
It follows that

(1110) 
Now, it is readily demonstrated that

(1111) 
so

(1112) 
Thus, our previous expressions for the transition rates for radiation induced absorption and
stimulated emission reduce to
respectively. Here,

(1115) 
is the effective electric dipole moment of the atom when making a
transition from state to state .
Equations (1113) and (1114) give the transition rates
for absorption and stimulated emission, respectively, induced by
a linearly polarized planewave. Actually, we are more interested in the
transition rates induced by unpolarized isotropic radiation. To obtain
these we must average Eqs. (1113) and (1114)
over all possible polarizations and propagation directions of the wave.
To facilitate this process, we can define a set of Cartesian coordinates
such that the wavevector , which specifies the direction
of wave propagation, points along the axis, and the vector ,
which specifies the direction of the atomic dipole moment, lies
in the  plane. It follows that the vector
, which
specifies the direction of wave polarization, must lie
in the  plane, since it has to be orthogonal to .
Thus, we can write
which implies that

(1119) 
We must now average the above quantity over all possible values of
and . Thus,

(1120) 
where
, and the integral is taken over all
solid angle. It is easily demonstrated that

(1121) 
Here, stands for

(1122) 
Hence, the transition rates
for absorption and stimulated emission induced by
unpolarized isotropic radiation are
respectively.
Next: Spontaneous Emission
Up: TimeDependent Perturbation Theory
Previous: Electromagnetic Radiation
Richard Fitzpatrick
20100720